The Lane-Emden conjecture for elliptic systems, with its Henon-weighted extension

Claims Theorem 1.1: for every $n\ge2$, $p,q>0$ and $A,B\in\mathbb R$ with $\frac{n+A}{p+1}+\frac{n+B}{q+1}>n-2$, the Henon-Lane-Emden system has no positive solution continuous at the origin and $C^2$ elsewhere, with no condition at infinity. Corollary 1.2 is the unweighted Lane-Emden conjecture; with Bidaut-Veron and Giacomini's radial existence theorem this gives Phan's Conjecture C in full for $n\ge3$, $A,B>-2$, and Huang-Zou's Conjecture B. It does NOT classify solutions on the critical hyperbola beyond the cited radial existence, and treats only positive (not sign-changing) solutions.

Claims Theorem 1.1: for every $n\ge2$, $p,q>0$ and $A,B\in\mathbb R$ with $\frac{n+A}{p+1}+\frac{n+B}{q+1}>n-2$, the Henon-Lane-Emden system has no positive solution continuous at the origin and $C^2$ elsewhere, with no condition at infinity. Corollary 1.2 is the unweighted Lane-Emden conjecture; with Bidaut-Veron and Giacomini's radial existence theorem this gives Phan's Conjecture C in full for $n\ge3$, $A,B>-2$, and Huang-Zou's Conjecture B. It does NOT classify solutions on the critical hyperbola beyond the cited radial existence, and treats only positive (not sign-changing) solutions.